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1 LIBRARY Technical REPOPTT SCOTtO* NAVAL POSTGRADUATE SCHQOff MONTERE*, CAUVOUtlA fiQ 77* * V/ / /O ^ (T Prepared for Office of Naval Research Contract No. N00014-69-A-0200-4048 A li *" t/ UCyVENG^414 FEBRUARY 1974 OPTIMUM DESIGN OF RING STIFFENED CYLINDRICAL SHELLS , BRONOWICKI FELTON SCHMIT. JR. Reproduction in whole or in part is permitted for any purpose of the United States Government. UCLA SCHOOL OF ENGINEERING AND APPLIED SCIENCE SECURITY CLASSIFICATION OF THIS PAGE (When Data Entered) REPORT DOCUMENTATION PAGE READ INSTRUCTIONS BEFORE COMPLETING FORM 1. REPORT NUMBER 2. GOVT ACCESSION NO 3. RECIPIENT'S CATALOG NUMBER 4.)

2 TITLE (and Subtitle) OPTIMUM DESIGN OF RING STIFFENED CYLINDRICAL SHELLS 5. TYPE OF REPORT & PERIOD COVERED 6. PERFORMING ORG. REPORT NUMBER UCLA-ENG-7414 7. AUTHORfsj Allen J. Bronowicki Richard B. Nelson Lewis P. Felton Lucien A. Schmit, Jr. 8 CONTRACT OR GRANT NUMBERfsJ Contract No. N00014-69-A-0200-4048 9. PERFORMING ORGANIZATION NAME AND ADDRESS Mechanics and Structures Department j School of Engineering and Applied Science j University of California, Los Angeles, Cal. 90024 T'lO. PROGRAM ELEMENT. PROJECT. TASK ! AREA A WORK UNIT NUMBERS II. CONTROLLING OFFICE NAME AND ADDRESS Procuring Contracting Officer Office of Naval Research Department of the Navy.

3 Arlington, Va. 12. REPORT DATE February 1974 22217 13. NUMBTR OF PAGES _22 U. MONITORING AGENCY NAME ft ADDRESS -'/ dlttarent from Controlling ) Director, Office of Naval Research Branch Office- Pasadena 1030 East Green Street Pasadena, Calif. 91101 15. SECURITY CLASS, (of this report) 15 DECl. ASSIFiCATION DOWNGRADING SOKC 16. DISTRIBUTION STATEMENT (of this Report; 17. DISTRIBUTION STATEMENT (o> tho abstract entered In Ulcck 20. It different from Report) 18. SUPPLEMENTARY NOTES 19. KEY WORDS (Continue on reverse aide if necessary and Identity by block number) Structural design, optimal; Shells, cylindrical; Vibration; Buckling 20.

4 ABSTRACT (Continue on reverae aide it neceaaary and Identity by block number) This report deals with the optimum structural design of circular cylindrical shells reinforced with identical uniformly spaced T-ring stiffeners, and sub- jected to external pressure loading. The optimization problems considered are of three types: (1) minimum-weight designs, (2) design for maximum separation of the lowest two natural frequencies, and (3) design for maximum separation of the lowest two natural frequencies which have primarily axial content. Gross buckling is precluded by specifying a minimum natural frequency, and DD 1 jAN 73 1473 EDITION OF I NOV 65 IS OBSOLETE S/N 0102-014-6601 SECURITY CLASSIFICATION OF THIS PAGE (When Dmtm Bntmrud) -LUURITY CLASSIFICATION OF THIS PAGECHTjen Data Entered) 20.

5 Additional behavioral constraints preclude yielding or buckling of panels, T-ring stiffeners, and web and flange instabilities within each T-ring. Analysis is based on use of an equivalent orthotropic shell model, and optimization is accomplished through use of a sequential unconstrained minimization technique. Examples indicate that a small increase in weight above optimum (minimum) values can result in rela- tively large increases in frequency separation, and that maximum fre- quency separation is obtained when second and third lowest frequencies approach each other. SECURITY CLASSIFICATION OF THIS PA0E(TW> n Data Entered) UCLA-ENG-7414 FEBRUARY 1974 OPTIMUM DESIGN OF RING STIFFENED CYLINDRICAL SHELLS Allen J.

6 Bronowicki Richard B. Nelson Lewis P. Felton Lucien A. Schmit, Jr. Prepared for Office of Naval Research Contract No. N00014-69-A-0200-4048 Mechanics and Structures Department School of Engineering and Applied Science University of California Los Angeles, California 90024 ACKNOWLEDGMENT The research described in this report was performed under Office of Naval Research Contract No. N00014-69-A-0200-4048. ii ABSTRACT This report deals with the optimum structural design of circular cylin- drical shells reinforced with identical uniformly spaced T-ring stiffeners, and subjected to external pressure loading. The optimization problems con- sidered are of three types: (1) minimum-weight design, (2) design for maximum separation of the lowest two natural frequencies, and (3) design for maximum separation of the lowest two natural frequencies which have primarily axial content.

7 Gross buckling is precluded by specifying a minimum natural fre- quency, and additional behavioral constraints preclude yielding or buckling of panels, T-ring stiffeners, and web and flange instabilities within each T-ring. Analysis is based on use of an equivalent orthotropic shell model, and optimiza- tion is accomplished through use of a sequential unconstrained minimization technique. Examples indicate that a small increase in weight above optimum (minimum) values can result in relatively large increases in frequency separation, and that maximum frequency separation is obtained when second and third lowest frequencies approach each other. iii SYMBOLS t Thickness of shell skin (in) s t.

8 Thickness of frame web (in) Thickness of frame flange (in) d. Depth of frame web (in) cL Width of frame flange (in) I Spacing of frames (in) W Normalized mass of structure 3 p Mass density of structural material (slug/in ) L Total length of structure (in) R Radius to mid-surface of skin (in) P Hydrostatic pressure (psi) T Kinetic energy x,<j),z Shell coordinate system u,v,w Midsurface displacements u,v,w Displacements of an arbitrary point {A}J Eigenvector of ith natural frequency 0) Natural frequency n,m Wave numbers T Time (sec) [K] Stiffness matrix [KQ] Geometric stiffness matrix [M]. Mass matrix a Yield stress (psi) a* Critical flange buckling stress (psi) cr aw Critical web buckling stress (psi) cr F Objective function iv X Vector of design variables g(x) Design constraint q Number of design variables p cr Critical pressure for buckling of skin (psi) between frames min Minimum allowable natural frequency of vibration in vacuo (Hz) E Young's modulus V Poisson's ratio & Unsupported length of shell plating (in) W Maximum allowable normalized structural mass max $(x)

9 Composite objective function e Constant for use in computing extended penalty function r Positive scalar quantity s Direction vector for uni-directional search INTRODUCTION Although a wealth of literature exists for the static, dynamic and sta- bility analyses of stiffened shells of revolution subjected to various applied loads, with the majority of these studies devoted to cylindrical shells, the work of Schmit and Morrow [1] serves as a pioneering effort toward the intro- duction of structural optimization concepts into the design of stiffened cylin- drical shells. In this reference a cylindrical shell, reinforced with longi- tudinal and ring stiffeners, each with rectangular cross section, was designed to carry a number of independently applied sets of static loads with minimum structural weight.

10 The shell was constrained against overall (system) buckling, panel and stiffener buckling, and also against material yield. The mathematical model which formed the basis for the stress and buckling analyses was an equiv- alent homogeneous orthotropic shell; , the discrete stiffeners and skin stiffness properties were incorporated in the orthotropic elastic shell stiff- ness properties. This theory, a 3rd order Flugge-Lur'e-Byrne type theory, proved adequate provided the stiffener spacing and cross sectional dimensions were sufficiently small to permit the smoothing operation inherent in the orthotropic shell model. In a more recent study, Pappas and Allentuch [2,3] investigated the mini- mum-weight design of ring stiffened cylindrical shells, subjected to a number of static applied load conditions.